production function

production function

a function that shows for a given state of technological knowledge the relationship between physical quantities of FACTOR INPUTS and the physical quantities of OUTPUT involved in producing a good or service. Since the quantity of output depends upon the quantities of inputs used, the relationship can be depicted in the form of FUNCTIONAL NOTATION:

where Q = output of a product and I1 I2, etc., are quantities of the various factor inputs 1, 2, etc., used in producing that output.

It is important to emphasize that factor inputs can be combined in a number of different ways to produce the same amount of output. One method may employ large amounts of capital and only small amounts of labour, while another method may employ large quantities of labour and only a little capital. In physical terms, the method that is technically the most efficient is the one that uses the fewest inputs. Economists, however, are more concerned with the cost aspects of the input-output relationship (see COST FUNCTION), specifically the least costly way of producing a given output. Such relationships can be analysed using ISOQUANT CURVES, ISOQUANT MAPS and PROCESS RAYS.

Notice that an inflection point [k.sub.f] exists such that f([k.sub.t]) is convex for [k.sub.t] [member of] (0, [k.sub.f]) and it is concave for k [member of] ([k.sub.f], +[infinity]) with [k.sub.f] = [(([rho] - 1)/[beta]([rho] + 1)).sup.1/[rho]], so it is a convex-concave production function. Such a function can be considered an extension of the Cobb-Douglas (CD) production function, since it approximates the CD case when [alpha] = 1 and [beta] = 0.

While the efficiency limit is considered as 1 in the Data Envelopment Analysis, in the Stochastic Frontier Analysis, it is determined based on production function. So, the highest result is not 1 in the Stochastic Frontier method.

Those in the first group (located at the left side of the tree diagram) are designed to estimate a production function (or cost function) and the efficiency of individual firms, whereas SFMs in the second group (located at the right side of the diagram) are useful for the analysis of efficiency factors even though these models can be also used for efficiency estimation.

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